You want to buy a house and wish to borrow $300,000. What would the monthly payment be if the loan requires equal monthly payments for 30 years at an interest rate of 5% APR? What would your outstanding loan balance be immediately after you make your first payment?

No, as long as the monthly loan payment is more than enough to cover the monthly interest accruing on the loan, you know that the loan's balance (principal) will diminish with each payment. Let's break this one down into bite-sized pieces:

Since you're given the three pieces of info: • original loan = 300K; • annual interest rate = 5%; and • loan's term is to be 30 years;

To deal with the second question, just step through the events of the first month:
• The loan will accrue interest for one month, which will be 300,000 x 0.05 ÷ 12.
• At the end of that first month, the borrower will make his first payment of $1,610.47, as determined previously.
• Of this payment to the bank, part of it will be used to pay the first month's interest, which you calculated at my first bullet point above.
• All of the rest of the 1,610.47---that is, the portion not being used to pay in full the first month's interest---will reduce the loan balance.
• Hence, the loan balance after the first monthly payment will be 300,000 less whatever principal reduction was obtained from that first payment.

Step through those points and fill in the blanks as you go.

Here's an equivalent way of viewing that second question: Imagine you're looking at a typical amortization schedule for this particular loan. Reading across the first line, you have the total loan payment made, the amount thereof applied to interest, the amount applied to principal, and finally, the loan balance after being reduced by the principal component of that first payment (obviously, it'll be a bit less than 300,000, but only a bit less, since most of that first monthly payment went towards interest).

That figure from the amortization schedule---that "little bit less than 300K" remaining balance---is what the second part of your question is looking for.

No, as long as the monthly loan payment is more than enough to cover the monthly interest accruing on the loan, you know that the loan's balance (principal) will diminish with each payment. Let's break this one down into bite-sized pieces:

Since you're given the three pieces of info: • original loan = 300K; • annual interest rate = 5%; and • loan's term is to be 30 years;

To deal with the second question, just step through the events of the first month:
• The loan will accrue interest for one month, which will be 300,000 x 0.05 ÷ 12.
• At the end of that first month, the borrower will make his first payment of $1,610.47, as determined previously.
• Of this payment to the bank, part of it will be used to pay the first month's interest, which you calculated at my first bullet point above.
• All of the rest of the 1,610.47---that is, the portion not being used to pay in full the first month's interest---will reduce the loan balance.
• Hence, the loan balance after the first monthly payment will be 300,000 less whatever principal reduction was obtained from that first payment.

Step through those points and fill in the blanks as you go.

Here's an equivalent way of viewing that second question: Imagine you're looking at a typical amortization schedule for this particular loan. Reading across the first line, you have the total loan payment made, the amount thereof applied to interest, the amount applied to principal, and finally, the loan balance after being reduced by the principal component of that first payment (obviously, it'll be a bit less than 300,000, but only a bit less, since most of that first monthly payment went towards interest).

That figure from the amortization schedule---that "little bit less than 300K" remaining balance---is what the second part of your question is looking for.